A Sobolev space approach for global solutions to certain semi-linear heat equations in bounded domains

Abstract

We present a Sobolev space approach for semilinear heat equations $u_t=\Delta u + F(u(t,x))$ for $t>0$ on a bounded domain $\Omega\subset\mathbf{R}^n$. By proving that there exists a solution in the anisotropic Sobolev space $W^{1,2}_p( \R_+\times\Omega)$, we can deduce more than just global existence in time. For example, both the solution and its time derivative are of class $L^p$, and the solution tends to zero in $L^\infty(\Omega)$ as $t\to\infty$. The main result shows that the existence of a solution in $W^{1,2}_p$ depends primarily on the existence of an appropriate a priori estimate on the $L^\infty$ norm of solutions as the initial data is deformed to zero.